The Fukui matrix : a simple approach to the analysis of the Fukui function and its positive character

The Fukui matrix is introduced as the derivative of the one-electron reduced density matrix with respect to a change in the number of electrons under constant external potential. The Fukui matrix extends the Fukui function concept: the diagonal of the Fukui matrix is the Fukui function. Diagonalizing the Fukui matrix gives a set of eigenvectors, the Fukui orbitals, and accompanying eigenvalues. At the level of theory used, there is always one dominant eigenvector, with an eigenvalue equal to 1. The remaining eigenvalues are either zero or come in pairs with eigenvalues of the same magnitude but opposite sign. Analysis of the frontier molecular orbital coefficient in the eigenvector with eigenvalue 1 gives information on the quality of the frontier molecular orbital picture. The occurrence of negative Fukui functions can be easily interpreted in terms of the nodal character of the dominant eigenvector versus the characteristics of the remaining eigenvectors and eigenvalues

Molecular quantum similarity measures in Minkowski metric vector semispaces

Minkowski metric vector semispaces can be chosen as the natural mathematical framework, where quantum similarity measures are described and evaluated. The obtained results in this study show that the Minkowski metric option is easily feasible, providing a new set of computationally simpler expressions, computationally faster when compared with Euclidian based quantum similarity measures

Mathematical elements of quantum electronic density functions

This chapter is a discussion on the electronic density functions formal structure and mathematical properties. A primary objective of this study is focused on the easy description of the quantum object concept, in connection with the quantum similarity measures framework. Several mathematical tools are discussed concerning the development of this task, among others: inward matrix products, extended Hilbert and Sobolev spaces, convex sets, vector semispaces, generating rules, diagonal representations, etc